Game Theory Lecture 9.

Slides:



Advertisements
Similar presentations
Game Theoretic Analysis of Oligopoly y n Y N 0000 Y N The unique dominant strategy Nash Equilibrium is (y,Y) A game of imperfect.
Advertisements

Vincent Conitzer CPS Repeated games Vincent Conitzer
Infinitely Repeated Games
Nash’s Theorem Theorem (Nash, 1951): Every finite game (finite number of players, finite number of pure strategies) has at least one mixed-strategy Nash.
Crime, Punishment, and Forgiveness
© 2009 Institute of Information Management National Chiao Tung University Game theory The study of multiperson decisions Four types of games Static games.
Evolution and Repeated Games D. Fudenberg (Harvard) E. Maskin (IAS, Princeton)
Simultaneous- Move Games with Mixed Strategies Zero-sum Games.
Managerial Economics Game Theory for Oligopoly
Infinitely Repeated Games. In an infinitely repeated game, the application of subgame perfection is different - after any possible history, the continuation.
Non-Cooperative Game Theory To define a game, you need to know three things: –The set of players –The strategy sets of the players (i.e., the actions they.
Chapter 14 Infinite Horizon 1.Markov Games 2.Markov Solutions 3.Infinite Horizon Repeated Games 4.Trigger Strategy Solutions 5.Investing in Strategic Capital.
M9302 Mathematical Models in Economics Instructor: Georgi Burlakov 2.5.Repeated Games Lecture
MIT and James Orlin © Game Theory 2-person 0-sum (or constant sum) game theory 2-person game theory (e.g., prisoner’s dilemma)
Infinitely Repeated Games Econ 171. Finitely Repeated Game Take any game play it, then play it again, for a specified number of times. The game that is.
Repeated games with Costly Observations Eilon Solan, Tel Aviv University Ehud Lehrer Tel Aviv University with.
EC941 - Game Theory Lecture 7 Prof. Francesco Squintani
Game Theory: Inside Oligopoly
Industrial Organization - Matilde Machado Tacit Collusion Tacit Collusion Matilde Machado.
Game Theory Lecture 8.
Games People Play. 8: The Prisoners’ Dilemma and repeated games In this section we shall learn How repeated play of a game opens up many new strategic.
Dynamic Games of Complete Information.. Repeated games Best understood class of dynamic games Past play cannot influence feasible actions or payoff functions.
A camper awakens to the growl of a hungry bear and sees his friend putting on a pair of running shoes, “You can’t outrun a bear,” scoffs the camper. His.
Yale Lectures 21 and Repeated Games: Cooperation vs the End Game.
ECON6036 1st semester Format of final exam Same as the mid term
UNIT II: The Basic Theory Zero-sum Games Nonzero-sum Games Nash Equilibrium: Properties and Problems Bargaining Games Review Midterm3/19 3/12.
APEC 8205: Applied Game Theory Fall 2007
Repeated games - example This stage game is played 2 times Any SPNE where players behave differently than in a 1-time game? Player 2 LMR L1, 10, 05, 0.
TOPIC 6 REPEATED GAMES The same players play the same game G period after period. Before playing in one period they perfectly observe the actions chosen.
UNIT II: The Basic Theory Zero-sum Games Nonzero-sum Games Nash Equilibrium: Properties and Problems Bargaining Games Bargaining and Negotiation Review.
Lecture 3: Oligopoly and Strategic Behavior Few Firms in the Market: Each aware of others’ actions Each firm in the industry has market power Entry is.
UNIT III: COMPETITIVE STRATEGY Monopoly Oligopoly Strategic Behavior 7/21.
0 MBA 299 – Section Notes 4/25/03 Haas School of Business, UC Berkeley Rawley.
EC941 - Game Theory Francesco Squintani Lecture 3 1.
UNIT II: The Basic Theory Zero-sum Games Nonzero-sum Games Nash Equilibrium: Properties and Problems Bargaining Games Bargaining and Negotiation Review.
Communication Networks A Second Course Jean Walrand Department of EECS University of California at Berkeley.
© 2009 Institute of Information Management National Chiao Tung University Lecture Notes II-2 Dynamic Games of Complete Information Extensive Form Representation.
1. problem set 6 from Osborne’s Introd. To G.T. p.210 Ex p.234 Ex p.337 Ex. 26,27 from Binmore’s Fun and Games.
C H A P T E R 13 Game Theory and Competitive Strategy CHAPTER OUTLINE
1 Game Theory Sequential bargaining and Repeated Games Univ. Prof.dr. M.C.W. Janssen University of Vienna Winter semester Week 46 (November 14-15)
Punishment and Forgiveness in Repeated Games. A review of present values.
3.1. Strategic Behavior Matilde Machado.
ECO290E: Game Theory Lecture 12 Static Games of Incomplete Information.
Dynamic Games of complete information: Backward Induction and Subgame perfection - Repeated Games -
Games with Imperfect Information Bayesian Games. Complete versus Incomplete Information So far we have assumed that players hold the correct belief about.
1 Topic 2 (continuation): Oligopoly Juan A. Mañez.
Chapters 29, 30 Game Theory A good time to talk about game theory since we have actually seen some types of equilibria last time. Game theory is concerned.
Lecture 5 Leadership and Reputation Reputations arise in situations where there is an element of repetition, and also where coordination between players.
Mixed Strategies and Repeated Games
Oligopoly Theory1 Oligopoly Theory (11) Collusion Aim of this lecture (1) To understand the idea of repeated game. (2) To understand the idea of the stability.
Lecture 6 Reputation Reputations arise in situations where there is an element of repetition, and also where coordination between players is possible.
Chapter 6 Extensive Form Games With Perfect Information (Illustrations)
Game Theory (Microeconomic Theory (IV)) Instructor: Yongqin Wang School of Economics, Fudan University December, 2004.
Extensive Form Games With Perfect Information (Illustrations)
Punishment, Detection, and Forgiveness in Repeated Games.
“Group consumption, free-riding, & informal reciprocity agreements”. Why do people use informal reciprocity agreements? Most analysis answers this question.
Lec 23 Chapter 28 Game Theory.
Econ 805 Advanced Micro Theory 1 Dan Quint Fall 2009 Lecture 1 A Quick Review of Game Theory and, in particular, Bayesian Games.
Entry Deterrence Players Two firms, entrant and incumbent Order of play Entrant decides to enter or stay out. If entrant enters, incumbent decides to fight.
1 Strategic Thinking Lecture 7: Repeated Strategic Situations Suggested reading: Dixit and Skeath, ch. 9 University of East Anglia School of Economics.
Game theory basics A Game describes situations of strategic interaction, where the payoff for one agent depends on its own actions as well as on the actions.
Dynamic Games of Complete Information
Vincent Conitzer CPS Repeated games Vincent Conitzer
Multiagent Systems Repeated Games © Manfred Huber 2018.
Vincent Conitzer Repeated games Vincent Conitzer
Chapter 14 & 15 Repeated Games.
Chapter 14 & 15 Repeated Games.
Molly W. Dahl Georgetown University Econ 101 – Spring 2009
Vincent Conitzer CPS Repeated games Vincent Conitzer
Presentation transcript:

Game Theory Lecture 9

problem set 9 from Osborne’s Introd. To G.T. Ex. 459.1, 459.2, 459.3

For a general game G: Repeated Games (a general treatment) C D What is the minimum that a player can guarantee? In the Prisoners’ Dilemma it was the payoff of (D,D) By playing D, player 2 can ensure that player 1 does not get more than 1 C D 2 , 2 0 , 3 3 , 0 1 , 1 For a general game G: Player 1 can always play the best response to the other’s action

Repeated Games (a general treatment) C D Player 2 can minimize the best that 1 can do by choosing t: C D 2 , 2 0 , 3 3 , 0 1 , 1 In the P.D. : In the P.D. it is a Nash equilibrium for each to play the stratgy that minimaxes the other. 3 1 In general playing the strategy that holds the other to his minimax payoff is NOT a Nash Equilibrium

Repeated Games (a general treatment) In the infinitely repeated game of G, every Nash equilibrium payoff is at least the minimax payoff If a player always plays the best response to his opponent’s action, his payoff is at least his minmax value. A folk theorem: approximately Every feasible value of G, which gives each player at least his minimax value, can be obtained as a Nash Equilibrium payoff (for δ~1)

? Repeated Games (a general treatment) If a point A is feasible, it can be (approximately) obtained by playing a cycle of actions. Consider a the following strategy: follow the cycle sequence if the sequence has been played in the past. If there was a deviation from it, play forever the action that holds the other to his minimax It is an equilibrium for both to play this strategy ?

Repeated Games (a general treatment) follow the cycle sequence if the sequence has been played in the past. If there was a deviation from it, play forever the action that holds the other to his minimax If both follow the strategy, each receives more than his minimax If one of them deviates, the other punishes him, hence the deviator gets at most his minimax. hence, he will not deviate.

Playing these strategies is Nash but not sub-game perfect equilibrium. Repeated Games (a general treatment) Would a player want to punish after a deviation??? by punishing the other his own payoff is reduced Playing these strategies is Nash but not sub-game perfect equilibrium. To make punishment ‘attractive’: it should last finitely many periods if not all participated in the punishment the counting starts again.

Repeated Games (a general treatment) An Example A B C 4 , 4 3 , 0 1 , 0 0 , 3 2 , 2 0 , 1 0 , 0 4 3 1 To ‘minimax’ the other one should play C when both play C, each gets 0

B C Repeated Games (a general treatment) A B C An Example 4 , 4 3 , 0 1 , 0 0 , 3 2 , 2 0 , 1 0 , 0 An Example a sub-game perfect equilibrium strategy: not (C,C) B C (B,B) not (B,B) (C,C) all 1 2 k .........

Incomplete Monitoring Two firms repeatedly compete in prices à la Bertrand, δ the discount rate Each observes its own profit but not the price set by the other. Demand is 0 with probability ρ, and D(p) with probability 1- ρ Assume that production unit cost is c, that D(p)0, and that (p-c)D(p) has a unique maximum at pm When demand is D(p), and both firms charge pm, each earns ½πm =½ (pm-c)D(pm).

Incomplete Monitoring Can the firms achieve cooperation (pm) ??? Let both firms play the following strategy (Sk): All pm c ½πm zero profit All 1 2 3 k For which values of k is the pair (Sk, Sk) a sub-game perfect equilibrium ???

Incomplete Monitoring Let V0 , V1 be the expected discouned payoffs at states 0,1 (respectively), when both players play Sk. 1 2 3 k pm c ½πm zero profit All

Incomplete Monitoring By the One Deviation Property, it suffices to check whether a deviation at state 0 can improve payoff. (At states 1,2,..k a deviation will not increase payoff). The best one can do at state 0, is to slightly undercut the other, this will yield a payoff of: 1 2 3 k pm c ½πm zero profit All

Incomplete Monitoring

…. …. Social Contract Overlapping Generations A person lives for 2 periods …. young old young old young old ….

Social Contract A young person produces 2 units of perishable good. An old person produces 0 units. A person’s preference for consumption over time (c1, c2), is given by: (1,1)  (2,0) It is an equilibrium for each young person to consume the 2 units he produces she produces Is there a ‘better’ equilibrium ??

Social Contract Let each young person give 1 unit to her old mother, provided the latter has, in her youth, given 1 unit to her own mother If my mother was ‘bad’ I am required to punish her, but then I will be punished in my old age. It is better not to follow this strategy.

This is a sub-game perfect equilibrium: Social Contract Let each young person give 1 unit to her old mother, provided ALL young persons in the past have contributed to their mothers. This is a sub-game perfect equilibrium: I am willing to punish my ‘bad’ mother, since I will be punished anyway.

more subtle strategies: Social Contract more subtle strategies: Punish your mother iff she is ‘bad’ A person is ‘bad’ if, either She did not provide her mother, although the mother was not ‘bad’. or: She did not punish her mother, although the mother was ‘bad’.

Incomplete Information meet avoid probability ½ probability ½ B X 2 , 1 0 , 0 1 , 2 B X 2 1 B X 2 1 B X 2 , 0 0 , 2 0 , 1 1 , 0

Incomplete Information ½ ½ B X 2 , 1 0 , 0 1 , 2 B X 2 , 0 0 , 2 0 , 1 1 , 0 meet avoid

½ ½ meet avoid 1 1 B X B X 2 2 2 2 B X B X B X B X